Some function spaces on symmetric spaces related to convolution operators
type: Item
instance of: scholarly article
title: Some function spaces on symmetric spaces related to convolution operators
zbMATH Open document ID: 0556.43002
DOI: 10.1016/0022-1236(84)90010-7
published in: Journal of Functional Analysis
publication date: 1984-01-01T00:00:00Z
review text: Let \(\mu\) be a positive measure on a semi-simple Lie group G equiped with a Haar measure. The authors obtain a necessary and sufficient condition on \(\mu\) such that it defines a bounded operator on \(L^ p(G)\) by convolutions under the assumption that \(\mu\) is invariant with respect to the action of the maximal compact subgroup K. By considering \(\mu\) as a measure on a symmetric space \(X=G/K\), the authors obtain a theorem similar to Kunz-Stein which states that if \(1\leq p<2\), every \(L^ p(G)\) function defines by convolution a bounded operator on \(L^ 2(G)\).
Mathematics Subject Classification ID: 43A15
zbMATH DE Number: 3886524
zbMATH Keywords: semi-simple Lie group
zbMATH Keywords: \(L^ p(G)\)
zbMATH Keywords: symmetric space
zbMATH Keywords: bounded operator
author: Noël Lohoué
author: Thomas Rychener
reviewed by: D. Somasundaram
MaRDI profile type: Publication
full work available at URL: 0022-1236(84)90010-7
OpenAlex ID: W2023573427
cites work: The dual of weak \(L^p\)
cites work: Sur la racine carrée du noyau de Poisson dans les espaces symétriques et une conjecture de E. M. Stein
cites work: Q5734920
cites work: A duality for symmetric spaces with applications to group representations
cites work: On L(p,q) spaces
cites work: Fatou Theorems for Eigenfunctions of the Invariant Differential Operators on Symmetric Spaces
cites work: Q5548784
cites work: Estimations \(L^ p\) des coefficients de représentation et opérateurs de convolution
cites work: Q3205762
cites work: Harmonic Functions on Hermitian Hyperbolic Space
cites work: Approach regions for the square root of the Poisson kernel and bounded functions
cites work: Q5640160
Recommended article: On Function Spaces on Symmetric Spaces
Recommended article: Q5286154
Recommended article: Convolution operators on spaces of entire functions
Recommended article: Symmetric functions on spaces $\ell_p(\mathbb{{R}}^n)$ and $\ell_p(\mathbb{{C}}^n)$
Recommended article: The convolution operation on the spectra of algebras of symmetric analytic functions
Recommended article: Convolution and involution on function spaces of homogeneous spaces
Recommended article: Q3806170
Recommended article: Q3725015
Recommended article: Convolution operators on spaces of holomorphic functions
Recommended article: SOME ASPECTS OF THE THEORY OF SYMMETRIC OPERATOR SPACES
description: scientific article; zbMATH DE number 3886524